Question

prove the 3 equations of motion
using calculus.

Answer 1

 


v²= u²+2as

Actually I don't know how to derive it. But here is some hints are provided.

You want it by integration.(v² - u²) /2 indicates the integration of v dv limit u to v.

Here is the proof -

a= dv/dt……1

v=ds/dt……….2

from 1 & 2:

a ds/dt = v dv/ dt

implies a ds =v dv

after integration you will get 3rd equation.

Answer 2

First equation of motion:

Consider a body having initial velocity u,If it is subjected to an acceleration 'a' so that after time 't' its velocity becomes v.

By the definition of acceleration

acceleration=Change in velocity/time taken

a=(v-u)/t

v=u+at

Second equation of motion:

Suppose the distance travelled by the above body in time t=s

we have average velocity=(Initial velocity+Final velocity)/2

average velocity =u+v/2
distance travelled =average velocity / time
s=v+u/2*t

using v=u+at in the above equation.
s=u+at+u/2*t
s=2ut+at²/2
s=ut+1/2at²

Third equation of motion:
using first equation of motion..
s=v+u*t/2
s=[v+u][v-u]/2a
s=v²-u²/2
2as=v²-u²
v²=u²+2as

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